Systematic secret sharing

ABSTRACT

A method includes obtaining an encoding matrix that includes a first matrix section and a second matrix section. The first matrix section includes “D−1” rows of a unity matrix, (“D” is a decode threshold number), and the second matrix section includes “T−D+1” rows of encoding terms (“T” is a pillar width number). The method further includes dividing a data element into “Z” data element blocks (“Z” is a function of “D” and a total number of data blocks of a data segment). The method further includes placing “Z” data element blocks in a first row of a data matrix that corresponds to a missing row of the unity matrix. The method further includes dividing “D−1” random elements into random element blocks and placing them in other rows of the data matrix. The method further includes matrix multiplying the encoding matrix with the data matrix to produce encoded data slices.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority pursuant to 35 U.S.C. § 120, as a continuation-in-part of U.S. Utility patent application Ser. No. 15/400,529, entitled “AUTOMATIC NAMESPACE ORDERING DETERMINATION,” filed Jan. 6, 2017, which claims priority pursuant to 35 U.S.C. § 120, as a continuation-in-part of U.S. Utility patent application Ser. No. 13/866,457, entitled “REPRIORITIZING PENDING DISPERSED STORAGE NETWORK REQUESTS,” filed Apr. 19, 2013, now U.S. Pat. No. 9,632,872, issued on Apr. 25, 2017, which claims priority pursuant to 35 U.S.C. § 119(e) to U.S. Provisional Application No. 61/655,753, entitled “ESTABLISHING AN ADDRESS RANGE ASSIGNMENT IN A DISTRIBUTED STORAGE AND TASK NETWORK,” filed Jun. 5, 2012, all of which are hereby incorporated herein by reference in their entirety and made part of the present U.S. Utility Patent Application for all purposes.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

Not Applicable.

INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC

Not Applicable.

BACKGROUND OF THE INVENTION Technical Field of the Invention

This invention relates generally to computer networks and more particularly to dispersing error encoded data.

Description of Related Art

Computing devices are known to communicate data, process data, and/or store data. Such computing devices range from wireless smart phones, laptops, tablets, personal computers (PC), work stations, and video game devices, to data centers that support millions of web searches, stock trades, or on-line purchases every day. In general, a computing device includes a central processing unit (CPU), a memory system, user input/output interfaces, peripheral device interfaces, and an interconnecting bus structure.

As is further known, a computer may effectively extend its CPU by using “cloud computing” to perform one or more computing functions (e.g., a service, an application, an algorithm, an arithmetic logic function, etc.) on behalf of the computer. Further, for large services, applications, and/or functions, cloud computing may be performed by multiple cloud computing resources in a distributed manner to improve the response time for completion of the service, application, and/or function. For example, Hadoop is an open source software framework that supports distributed applications enabling application execution by thousands of computers.

In addition to cloud computing, a computer may use “cloud storage” as part of its memory system. As is known, cloud storage enables a user, via its computer, to store files, applications, etc. on an Internet storage system. The Internet storage system may include a RAID (redundant array of independent disks) system and/or a dispersed storage system that uses an error correction scheme to encode data for storage.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

FIG. 1 is a schematic block diagram of an embodiment of a dispersed or distributed storage network (DSN) in accordance with the present invention;

FIG. 2 is a schematic block diagram of an embodiment of a computing core in accordance with the present invention;

FIG. 3 is a schematic block diagram of an example of dispersed storage error encoding of data in accordance with the present invention;

FIG. 4 is a schematic block diagram of a generic example of an error encoding function in accordance with the present invention;

FIG. 5 is a schematic block diagram of a specific example of an error encoding function in accordance with the present invention;

FIG. 6 is a schematic block diagram of an example of a slice name of an encoded data slice (EDS) in accordance with the present invention;

FIG. 7 is a schematic block diagram of an example of dispersed storage error decoding of data in accordance with the present invention;

FIG. 8 is a schematic block diagram of a generic example of an error decoding function in accordance with the present invention;

FIG. 9 is a diagram illustrating an example of matrix multiplication to encode a secret in accordance with the present invention;

FIG. 10 is a schematic block diagram of an example of matrix multiplication to encode a data element in accordance with the present invention;

FIG. 11 is a schematic block diagram of another example of matrix multiplication to encode a data element in accordance with the present invention;

FIG. 12 is a schematic block diagram of another example of matrix multiplication to encode a data element in accordance with the present invention; and

FIG. 13 is a logic diagram of an example of a method of matrix multiplication to encode a data element in accordance with the present invention.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 is a schematic block diagram of an embodiment of a dispersed, or distributed, storage network (DSN) 10 that includes a plurality of computing devices 12-16, a managing unit 18, an integrity processing unit 20, and a DSN memory 22. The components of the DSN 10 are coupled to a network 24, which may include one or more wireless and/or wire lined communication systems; one or more non-public intranet systems and/or public internet systems; and/or one or more local area networks (LAN) and/or wide area networks (WAN).

The DSN memory 22 includes a plurality of storage units 36 that may be located at geographically different sites (e.g., one in Chicago, one in Milwaukee, etc.), at a common site, or a combination thereof. For example, if the DSN memory 22 includes eight storage units 36, each storage unit is located at a different site. As another example, if the DSN memory 22 includes eight storage units 36, all eight storage units are located at the same site. As yet another example, if the DSN memory 22 includes eight storage units 36, a first pair of storage units are at a first common site, a second pair of storage units are at a second common site, a third pair of storage units are at a third common site, and a fourth pair of storage units are at a fourth common site. Note that a DSN memory 22 may include more or less than eight storage units 36. Further note that each storage unit 36 includes a computing core (as shown in FIG. 2, or components thereof) and a plurality of memory devices for storing dispersed error encoded data.

Each of the computing devices 12-16, the managing unit 18, and the integrity processing unit 20 include a computing core 26, which includes network interfaces 30-33. Computing devices 12-16 may each be a portable computing device and/or a fixed computing device. A portable computing device may be a social networking device, a gaming device, a cell phone, a smart phone, a digital assistant, a digital music player, a digital video player, a laptop computer, a handheld computer, a tablet, a video game controller, and/or any other portable device that includes a computing core. A fixed computing device may be a computer (PC), a computer server, a cable set-top box, a satellite receiver, a television set, a printer, a fax machine, home entertainment equipment, a video game console, and/or any type of home or office computing equipment. Note that each of the managing unit 18 and the integrity processing unit 20 may be separate computing devices, may be a common computing device, and/or may be integrated into one or more of the computing devices 12-16 and/or into one or more of the storage units 36.

Each interface 30, 32, and 33 includes software and hardware to support one or more communication links via the network 24 indirectly and/or directly. For example, interface 30 supports a communication link (e.g., wired, wireless, direct, via a LAN, via the network 24, etc.) between computing devices 14 and 16. As another example, interface 32 supports communication links (e.g., a wired connection, a wireless connection, a LAN connection, and/or any other type of connection to/from the network 24) between computing devices 12 & 16 and the DSN memory 22. As yet another example, interface 33 supports a communication link for each of the managing unit 18 and the integrity processing unit 20 to the network 24.

Computing devices 12 and 16 include a dispersed storage (DS) client module 34, which enables the computing device to dispersed storage error encode and decode data as subsequently described with reference to one or more of FIGS. 3-8. In this example embodiment, computing device 16 functions as a dispersed storage processing agent for computing device 14. In this role, computing device 16 dispersed storage error encodes and decodes data on behalf of computing device 14. With the use of dispersed storage error encoding and decoding, the DSN 10 is tolerant of a significant number of storage unit failures (the number of failures is based on parameters of the dispersed storage error encoding function) without loss of data and without the need for a redundant or backup copies of the data. Further, the DSN 10 stores data for an indefinite period of time without data loss and in a secure manner (e.g., the system is very resistant to unauthorized attempts at accessing the data).

In operation, the managing unit 18 performs DS management services. For example, the managing unit 18 establishes distributed data storage parameters (e.g., vault creation, distributed storage parameters, security parameters, billing information, user profile information, etc.) for computing devices 12-14 individually or as part of a group of user devices. As a specific example, the managing unit 18 coordinates creation of a vault (e.g., a virtual memory block associated with a portion of an overall namespace of the DSN) within the DSTN memory 22 for a user device, a group of devices, or for public access and establishes per vault dispersed storage (DS) error encoding parameters for a vault. The managing unit 18 facilitates storage of DS error encoding parameters for each vault by updating registry information of the DSN 10, where the registry information may be stored in the DSN memory 22, a computing device 12-16, the managing unit 18, and/or the integrity processing unit 20.

The DSN managing unit 18 creates and stores user profile information (e.g., an access control list (ACL)) in local memory and/or within memory of the DSN memory 22. The user profile information includes authentication information, permissions, and/or the security parameters. The security parameters may include encryption/decryption scheme, one or more encryption keys, key generation scheme, and/or data encoding/decoding scheme.

The DSN managing unit 18 creates billing information for a particular user, a user group, a vault access, public vault access, etc. For instance, the DSTN managing unit 18 tracks the number of times a user accesses a non-public vault and/or public vaults, which can be used to generate a per-access billing information. In another instance, the DSTN managing unit 18 tracks the amount of data stored and/or retrieved by a user device and/or a user group, which can be used to generate a per-data-amount billing information.

As another example, the managing unit 18 performs network operations, network administration, and/or network maintenance. Network operations includes authenticating user data allocation requests (e.g., read and/or write requests), managing creation of vaults, establishing authentication credentials for user devices, adding/deleting components (e.g., user devices, storage units, and/or computing devices with a DS client module 34) to/from the DSN 10, and/or establishing authentication credentials for the storage units 36. Network administration includes monitoring devices and/or units for failures, maintaining vault information, determining device and/or unit activation status, determining device and/or unit loading, and/or determining any other system level operation that affects the performance level of the DSN 10. Network maintenance includes facilitating replacing, upgrading, repairing, and/or expanding a device and/or unit of the DSN 10.

The integrity processing unit 20 performs rebuilding of ‘bad’ or missing encoded data slices. At a high level, the integrity processing unit 20 performs rebuilding by periodically attempting to retrieve/list encoded data slices, and/or slice names of the encoded data slices, from the DSN memory 22. For retrieved encoded slices, they are checked for errors due to data corruption, outdated version, etc. If a slice includes an error, it is flagged as a ‘bad’ slice. For encoded data slices that were not received and/or not listed, they are flagged as missing slices. Bad and/or missing slices are subsequently rebuilt using other retrieved encoded data slices that are deemed to be good slices to produce rebuilt slices. The rebuilt slices are stored in the DSTN memory 22.

FIG. 2 is a schematic block diagram of an embodiment of a computing core 26 that includes a processing module 50, a memory controller 52, main memory 54, a video graphics processing unit 55, an input/output (IO) controller 56, a peripheral component interconnect (PCI) interface 58, an IO interface module 60, at least one IO device interface module 62, a read only memory (ROM) basic input output system (BIOS) 64, and one or more memory interface modules. The one or more memory interface module(s) includes one or more of a universal serial bus (USB) interface module 66, a host bus adapter (HBA) interface module 68, a network interface module 70, a flash interface module 72, a hard drive interface module 74, and a DSN interface module 76.

The DSN interface module 76 functions to mimic a conventional operating system (OS) file system interface (e.g., network file system (NFS), flash file system (FFS), disk file system (DFS), file transfer protocol (FTP), web-based distributed authoring and versioning (WebDAV), etc.) and/or a block memory interface (e.g., small computer system interface (SCSI), internet small computer system interface (iSCSI), etc.). The DSN interface module 76 and/or the network interface module 70 may function as one or more of the interface 30-33 of FIG. 1. Note that the IO device interface module 62 and/or the memory interface modules 66-76 may be collectively or individually referred to as IO ports.

FIG. 3 is a schematic block diagram of an example of dispersed storage error encoding of data. When a computing device 12 or 16 has data to store it disperse storage error encodes the data in accordance with a dispersed storage error encoding process based on dispersed storage error encoding parameters. The dispersed storage error encoding parameters include an encoding function (e.g., information dispersal algorithm, Reed-Solomon, Cauchy Reed-Solomon, systematic encoding, non-systematic encoding, on-line codes, etc.), a data segmenting protocol (e.g., data segment size, fixed, variable, etc.), and per data segment encoding values. The per data segment encoding values include a total, or pillar width, number (T) of encoded data slices per encoding of a data segment i.e., in a set of encoded data slices); a decode threshold number (D) of encoded data slices of a set of encoded data slices that are needed to recover the data segment; a read threshold number (R) of encoded data slices to indicate a number of encoded data slices per set to be read from storage for decoding of the data segment; and/or a write threshold number (W) to indicate a number of encoded data slices per set that must be accurately stored before the encoded data segment is deemed to have been properly stored. The dispersed storage error encoding parameters may further include slicing information (e.g., the number of encoded data slices that will be created for each data segment) and/or slice security information (e.g., per encoded data slice encryption, compression, integrity checksum, etc.).

In the present example, Cauchy Reed-Solomon has been selected as the encoding function (a generic example is shown in FIG. 4 and a specific example is shown in FIG. 5); the data segmenting protocol is to divide the data object into fixed sized data segments; and the per data segment encoding values include: a pillar width of 5, a decode threshold of 3, a read threshold of 4, and a write threshold of 4. In accordance with the data segmenting protocol, the computing device 12 or 16 divides the data (e.g., a file (e.g., text, video, audio, etc.), a data object, or other data arrangement) into a plurality of fixed sized data segments (e.g., 1 through Y of a fixed size in range of Kilo-bytes to Tera-bytes or more). The number of data segments created is dependent of the size of the data and the data segmenting protocol.

The computing device 12 or 16 then disperse storage error encodes a data segment using the selected encoding function (e.g., Cauchy Reed-Solomon) to produce a set of encoded data slices. FIG. 4 illustrates a generic Cauchy Reed-Solomon encoding function, which includes an encoding matrix (EM), a data matrix (DM), and a coded matrix (CM). The size of the encoding matrix (EM) is dependent on the pillar width number (T) and the decode threshold number (D) of selected per data segment encoding values. To produce the data matrix (DM), the data segment is divided into a plurality of data blocks and the data blocks are arranged into D number of rows with Z data blocks per row. Note that Z is a function of the number of data blocks created from the data segment and the decode threshold number (D). The coded matrix is produced by matrix multiplying the data matrix by the encoding matrix.

FIG. 5 illustrates a specific example of Cauchy Reed-Solomon encoding with a pillar number (T) of five and decode threshold number of three. In this example, a first data segment is divided into twelve data blocks (D1-D12). The coded matrix includes five rows of coded data blocks, where the first row of X11-X14 corresponds to a first encoded data slice (EDS 1_1), the second row of X21-X24 corresponds to a second encoded data slice (EDS 2_1), the third row of X31-X34 corresponds to a third encoded data slice (EDS 3_1), the fourth row of X41-X44 corresponds to a fourth encoded data slice (EDS 4_1), and the fifth row of X51-X54 corresponds to a fifth encoded data slice (EDS 5_1). Note that the second number of the EDS designation corresponds to the data segment number.

Returning to the discussion of FIG. 3, the computing device also creates a slice name (SN) for each encoded data slice (EDS) in the set of encoded data slices. A typical format for a slice name 60 is shown in FIG. 6. As shown, the slice name (SN) 60 includes a pillar number of the encoded data slice (e.g., one of 1-T), a data segment number (e.g., one of 1-Y), a vault identifier (ID), a data object identifier (ID), and may further include revision level information of the encoded data slices. The slice name functions as, at least part of, a DSN address for the encoded data slice for storage and retrieval from the DSN memory 22.

As a result of encoding, the computing device 12 or 16 produces a plurality of sets of encoded data slices, which are provided with their respective slice names to the storage units for storage. As shown, the first set of encoded data slices includes EDS 1_1 through EDS 5_1 and the first set of slice names includes SN 1_1 through SN 5_1 and the last set of encoded data slices includes EDS 1_Y through EDS 5_Y and the last set of slice names includes SN 1_Y through SN 5_Y.

FIG. 7 is a schematic block diagram of an example of dispersed storage error decoding of a data object that was dispersed storage error encoded and stored in the example of FIG. 4. In this example, the computing device 12 or 16 retrieves from the storage units at least the decode threshold number of encoded data slices per data segment. As a specific example, the computing device retrieves a read threshold number of encoded data slices.

To recover a data segment from a decode threshold number of encoded data slices, the computing device uses a decoding function as shown in FIG. 8. As shown, the decoding function is essentially an inverse of the encoding function of FIG. 4. The coded matrix includes a decode threshold number of rows (e.g., three in this example) and the decoding matrix in an inversion of the encoding matrix that includes the corresponding rows of the coded matrix. For example, if the coded matrix includes rows 1, 2, and 4, the encoding matrix is reduced to rows 1, 2, and 4, and then inverted to produce the decoding matrix.

FIG. 9 is a diagram illustrating an example of matrix multiplication to encode a secret. An algebraic expression of a form f(n)=r₁x²+r₀x¹+sx⁰ is utilized to convey a secret (e.g., meaningful data) when storing or communicating at least a decode threshold number of variants of the algebraic expression (e.g., with different values for the variable x). The decode threshold number is the number of terms of the algebraic expression (e.g., 3). A width number of shares may be stored or communicated to include the at least the decode threshold number of variants of the algebraic expression to improve a reliability level of decoding the decode threshold number of variants of the algebraic expression to reproduce the secret. Coefficients of the terms of the algebraic expression include the secret (e.g., s) and a decode threshold minus one number of random coefficients (e.g., r₁ and r₀).

The width number of shares may be generated by matrix multiplying an encoding matrix (E) by a vector matrix (V). The vector matrix (V) includes generating a one column matrix that includes the decode threshold number of coefficients (e.g. s, r₁, r₀). For example, the secret s is assigned to a first row, coefficient r₀ is assigned to a second row, and coefficient r₁ is assigned to a third row of the vector matrix. The encoding matrix may be generated by generating a shortened identity matrix, generating a Vandermonde matrix, and combining the shortened identity matrix and the Vandermonde matrix to produce the encoding matrix. The shortened identity matrix may be generated by generating an identity matrix with a decode threshold number of rows and columns and deleting a row corresponding to a position of the secret in the vector matrix. For example, a 3×3 identity matrix is generated and a first row is deleted corresponding to the position of the secret in the vector matrix.

The Vandermonde matrix may be generated to include a width minus decode threshold number plus one number of rows and a decode threshold number of columns. Each row of the Vandermonde matrix includes a value for a variable of the algebraic expression, wherein each row includes a different value. The values are subsequently known to a decoding process. For example, a first row includes a value of a, a second row includes a value of b, and the third row includes a value of c. Each column of the Vandermonde matrix includes the value of the variable to a power represented in the algebraic expression. For example, a first column includes a value of the variable to the zero power, the second column includes a value of the variable to the first power, and the third column includes a value of the variable to the second power. In an example of matrix multiplying the encoding matrix by the vector matrix, share 1=0s+1r0+0r1=r0, share 2=0s+0r0+1r1=r1, share 3=sa⁰+r₀a¹+r₁a², share 4=sb⁰+r₀b¹+r₁b², and share 5=sc⁰+r₀c¹+r₁c².

Subsequent decoding of the secret includes retrieving the shares associated with the random coefficients and at least one other share, directly extracting the values of the random coefficients from the shares associated with the random coefficients, and solving an algebraic expression of the other share for the secret utilizing the values of the random coefficients. For example, decoding of the secret includes retrieving shares 1 and 2, extracting the random coefficients directly from shares 1 and 2, retrieving at least one share of shares 3-5, and solving and algebraic expression of the at least one share to reproduce the secret. As such, solving simultaneous linear equations is not required when a decode threshold number minus one number of random coefficients are directly available via the shares associated with the random coefficients and a decoding loading efficiency improvement is provided.

FIG. 10 is a schematic block diagram of an example of matrix multiplication to encode a data element. The example includes an encoding matrix (E) that is in accordance with a dispersed storage error encoding function. The encoding matrix (E) includes a first matrix section 88 and a second matrix section 90. The first matrix section 88 includes “D−1” (where “D” is the decode threshold number of the dispersed storage error encoding function) rows of a unity matrix. The second matrix section 90 includes “T−D+1” (where “T” is the pillar width number of the dispersed storage error encoding function) rows of encoding terms (e.g., coefficients or variables).

Here, the decode threshold number “D” is 3 and the pillar width number “T” is 5. Therefore, the first matrix section includes “D−1”=2 rows of a unity matrix where the first row of the unity matrix is missing in this example. The second matrix section 90 includes “T−D+1”=3 rows of encoding terms (e.g., encoding coefficients g-o).

A data matrix (DM) is created from a data segment 82. Data segment 82 includes a data element 84 and “D−1” (e.g., “D−1”=2) random elements (e.g., random element 1 and random element 2). The data element 84 includes meaningful data whereas random elements 1-2 include one or more of a pseudo random generated number, a known binary number pattern, and a random alpha numeric sequence.

Data element 84, random element 1, and random element 2 are divided into a “Z” number of blocks. “Z” is a function of the total number of blocks created from data segment 82 and the decode threshold number “D.” Here the total number of blocks created from data segment 82 is 12 and the decode threshold number “D” is 3. Therefore, the data element 84 is divided into “Z”=12/3=4 data elements blocks (e.g., DE1, DE2, DE3, D4). Random element 1 is divided into “Z”=4 random element blocks (e.g., RE1, RE2, RE3, and RE4). Random element 2 is divided into “Z”=4 random element blocks (e.g., RE5, RE6, RE7, and RE8).

The “Z” number of data element blocks DE1-DE4 is placed in a row of data matrix (DM) that corresponds to the missing row of the unity matrix. In this example, the first row of the unity matrix is missing so data element blocks DE1-DE4 are placed in the first row of the data matrix (DM). The random element blocks are placed in the other rows of the data matrix (DM). Matrix multiplying the encoding matrix (E) with the data matrix (DM) produces a coded matrix (C) which is translated into a set of encoded data slices (EDSs) 86.

Based on the placement of the data element blocks in the data matrix (DM), no data element block is present in any one slice of the set of encoded data slices 86. Data element 84 can be reconstructed by obtaining EDS 1_1 and EDS 2_1 along with at least one of EDS 3_1, EDS 4_1, and EDS 5_1. For example, obtaining EDS 1_1 (including RE1-RE4) and EDS 2_1 (including RE5-RE8) provides random element 1 and random element 2 which is enough information to decode the data element 84 from any of EDS 3_1, EDS 4_1, and EDS 5_1. Therefore, security is provided for data element 84 because no data element block is present in an individual slice. Further, decoding costs are reduced because only one slice needs to be decoded (e.g., at least one of EDS 3_1, EDS 4_1, and EDS 5_1) if EDS 1_1 and EDS 2_1 are obtained.

FIG. 11 is a schematic block diagram of another example of matrix multiplication to encode a data element. The example includes an encoding matrix (E) that is in accordance with a dispersed storage error encoding function. The encoding matrix (E) includes a first matrix section 88 and a second matrix section 90. The first matrix section 88 includes “D−1” (where “D” is the decode threshold number of the dispersed storage error encoding function) rows of a unity matrix. The second matrix section 90 includes “T−D+1” (where “T” is the pillar width number of the dispersed storage error encoding function) rows of encoding terms (e.g., coefficients or variables).

Here, the decode threshold number “D” is 3, and the pillar width number “T” is 5. Therefore, the first matrix section includes “D−1”=2 rows of a unity matrix where a second row of the unity matrix is missing in this example. The second matrix section 90 includes “T−D+1”=3 rows of encoding terms (e.g., encoding coefficients g-o).

A data matrix (DM) is created from a data segment 82. Data segment 82 includes a data element 84 and “D−1” (e.g., “D−1”=2) random elements (e.g., random element 1 and random element 2). The data element 84 includes meaningful data whereas random elements 1-2 include one or more of a pseudo random generated number, a known binary number pattern, and a random alpha numeric sequence.

Data element 84, random element 1, and random element 2 are divided into a “Z” number of blocks. “Z” is a function of the total number of blocks created from data segment 82 and the decode threshold number “D.” Here the total number of blocks created from data segment 82 is 12 and the decode threshold number “D” is 3. Therefore, the data element 84 is divided into “Z”=12/3=4 data elements blocks (e.g., DE1, DE2, DE3, D4). Random element 1 is divided into “Z”=4 random element blocks (e.g., RE1, RE2, RE3, and RE4). Random element 2 is divided into “Z”=4 random element blocks (e.g., RE5, RE6, RE7, and RE8).

The “Z” number of data element blocks DE1-DE4 is placed in a row of data matrix (DM) that corresponds to the missing row of the unity matrix. In this example, the second row of the unity matrix is missing so data element blocks DE1-DE4 are placed in the second row of the data matrix (DM). The random element blocks are placed in the other rows of the data matrix (DM). Matrix multiplying the encoding matrix (E) with the data matrix (DM) produces a coded matrix (C) which is translated into a set of encoded data slices (EDSs) 86.

As was true in FIG. 10, no data element block is present in any one slice of the set of encoded data slices 86. Data element 84 can be reconstructed by obtaining EDS 1_1 and EDS 2_1 along with at least one of EDS 3_1, EDS 4_1, and EDS 5_1. For example, obtaining EDS 1_1 (including RE1-RE4) and EDS 2_1 (including RE5-RE8) provides random element 1 and random element 2 which is enough information to decode the data element 84 from any of EDS 3_1, EDS 4_1, and EDS 5_1.

FIG. 12 is a schematic block diagram of another example of matrix multiplication to encode a data element. The example includes an encoding matrix (E) that is in accordance with a dispersed storage error encoding function. The encoding matrix (E) includes a first matrix section 88 and a second matrix section 90. The first matrix section 88 includes “D−1” (where “D” is the decode threshold number of the dispersed storage error encoding function) rows of a unity matrix. The second matrix section 90 includes “T−D+1” (where “T” is the pillar width number of the dispersed storage error encoding function) rows of a Vandermonde matrix where “D” is the number of terms of an algebraic expression of a form f(n)=r₁x²+r₀x¹+sx⁰.

Here, the decode threshold number “D” is 3, and the pillar width number “T” is 5. Therefore, the first matrix section includes “D−1”=2 rows of a unity matrix where a first row of the unity matrix is missing in this example. The second matrix section 90 includes “T−D+1”=3 rows of a Vandermonde matrix where the number of terms of the algebraic expression is 3.

A data matrix (DM) is created from a data segment 82. Data segment 82 includes a data element 84 and “D−1” (e.g., “D−1”=2) random elements (e.g., random element 1 and random element 2). The data element 84 includes meaningful data whereas random elements 1-2 include one or more of a pseudo random generated number, a known binary number pattern, and a random alpha numeric sequence.

Data element 84, random element 1, and random element 2 are divided into a “Z” number of blocks. “Z” is a function of the total number of blocks created from data segment 82 and the decode threshold number “D.” Here the total number of blocks created from data segment 82 is 12 and the decode threshold number “D” is 3. Therefore, the data element 84 is divided into “Z”=12/3=4 data elements blocks (e.g., DE1, DE2, DE3, D4). Random element 1 is divided into “Z”=4 random element blocks (e.g., RE1, RE2, RE3, and RE4). Random element 2 is divided into “Z”=4 random element blocks (e.g., RE5, RE6, RE7, and RE8). The data element blocks and random element blocks serve as coefficients of the terms of the algebraic expression.

The “Z” number of data element blocks DE1-DE4 is placed in a row of data matrix (DM) that corresponds to the missing row of the unity matrix. In this example, the first row of the unity matrix is missing so data element blocks DE1-DE4 are placed in the first row of the data matrix (DM). The random element blocks are placed in the other rows of the data matrix (DM). Matrix multiplying the encoding matrix (E) with the data matrix (DM) produces a coded matrix (C) which is translated into a set of encoded data slices (EDSs) 86.

As was true in FIGS. 10-11, no data element block is present in any one slice of the set of encoded data slices 86. Data element 84 can be reconstructed by obtaining EDS 1_1 and EDS 2_1, extracting the values of the random element block coefficients and solving the algebraic expressions of at least one of EDS 3_1, EDS 4_1, and EDS 5_1 utilizing the values of the random element block coefficients. For example, obtaining EDS 1_1 (including RE1-RE4) and EDS 2_1 (including RE5-RE8) provides random element 1 and random element 2 which is enough information to decode the data element 84 from any of EDS 3_1, EDS 4_1, and EDS 5_1. As such, solving simultaneous linear equations is not required.

FIG. 13 is a logic diagram of an example of a method of matrix multiplication to encode a data element. The method begins with step 92 where a computing device of a dispersed storage network (DSN) obtains an encoding matrix that includes a first matrix section and a second matrix section. The first matrix section includes “D−1” (where “D” is the decode threshold number of the dispersed storage error encoding function) rows of a unity matrix. The second matrix section includes “T−D+1” (where “T” is the pillar width number of the dispersed storage error encoding function) rows of encoding terms (e.g., coefficients or variables). For example, the second matrix section includes “T−D+1” rows of a Vandermonde matrix.

The method continues with step 94 where the computing device divides a data element into “Z” data element blocks (where “Z” is a function of “D” and a total number of data blocks of a data segment and the data segment includes the data element and “D−1” random elements). The data element includes meaningful data whereas a random element of the “D−1” random elements includes one or more of a pseudo random generated number, a known binary number pattern, and a random alpha numeric sequence.

The method continues with step 96 where the computing device places the “Z” data element blocks in a first row of the data matrix, where the first row corresponds to a missing row of the “D−1” number of rows of the unity matrix. For example, when a first row of the unity matrix is missing, the computing device places the “Z” data element blocks in a first row of the data matrix. As another example, when a second row of the unity matrix is missing, the computing device places the “Z” data element blocks in a second row of the data matrix. As another example, when a third row of the unity matrix is missing, the computing device places the “Z” data element blocks in a third row of the data matrix.

The method continues with step 98 where the computing device divides the “D−1” random elements into a plurality of random element blocks, where each of the “D−1” random elements is divided into “Z” random element blocks. The method continues with step 100 where the computing device places the plurality of random element blocks in other rows of the data matrix. The method continues with step 102 where the computing device matrix multiplies the encoding matrix with the data matrix to produce a set of encoded data slices. Based on the placement of the data element blocks in the data matrix, no data element block is present in any one slice of the set of encoded data slices. To reconstruct the data element, only the slices containing the “D−1” random elements plus one other slice of the set of encoded data slices need to be obtained.

It is noted that terminologies as may be used herein such as bit stream, stream, signal sequence, etc. (or their equivalents) have been used interchangeably to describe digital information whose content corresponds to any of a number of desired types (e.g., data, video, speech, audio, etc. any of which may generally be referred to as ‘data’).

As may be used herein, the terms “substantially” and “approximately” provides an industry-accepted tolerance for its corresponding term and/or relativity between items. Such an industry-accepted tolerance ranges from less than one percent to fifty percent and corresponds to, but is not limited to, component values, integrated circuit process variations, temperature variations, rise and fall times, and/or thermal noise. Such relativity between items ranges from a difference of a few percent to magnitude differences. As may also be used herein, the term(s) “configured to”, “operably coupled to”, “coupled to”, and/or “coupling” includes direct coupling between items and/or indirect coupling between items via an intervening item (e.g., an item includes, but is not limited to, a component, an element, a circuit, and/or a module) where, for an example of indirect coupling, the intervening item does not modify the information of a signal but may adjust its current level, voltage level, and/or power level. As may further be used herein, inferred coupling (i.e., where one element is coupled to another element by inference) includes direct and indirect coupling between two items in the same manner as “coupled to”. As may even further be used herein, the term “configured to”, “operable to”, “coupled to”, or “operably coupled to” indicates that an item includes one or more of power connections, input(s), output(s), etc., to perform, when activated, one or more its corresponding functions and may further include inferred coupling to one or more other items. As may still further be used herein, the term “associated with”, includes direct and/or indirect coupling of separate items and/or one item being embedded within another item.

As may be used herein, the term “compares favorably”, indicates that a comparison between two or more items, signals, etc., provides a desired relationship. For example, when the desired relationship is that signal 1 has a greater magnitude than signal 2, a favorable comparison may be achieved when the magnitude of signal 1 is greater than that of signal 2 or when the magnitude of signal 2 is less than that of signal 1. As may be used herein, the term “compares unfavorably”, indicates that a comparison between two or more items, signals, etc., fails to provide the desired relationship.

As may also be used herein, the terms “processing module”, “processing circuit”, “processor”, and/or “processing unit” may be a single processing device or a plurality of processing devices. Such a processing device may be a microprocessor, micro-controller, digital signal processor, microcomputer, central processing unit, field programmable gate array, programmable logic device, state machine, logic circuitry, analog circuitry, digital circuitry, and/or any device that manipulates signals (analog and/or digital) based on hard coding of the circuitry and/or operational instructions. The processing module, module, processing circuit, and/or processing unit may be, or further include, memory and/or an integrated memory element, which may be a single memory device, a plurality of memory devices, and/or embedded circuitry of another processing module, module, processing circuit, and/or processing unit. Such a memory device may be a read-only memory, random access memory, volatile memory, non-volatile memory, static memory, dynamic memory, flash memory, cache memory, and/or any device that stores digital information. Note that if the processing module, module, processing circuit, and/or processing unit includes more than one processing device, the processing devices may be centrally located (e.g., directly coupled together via a wired and/or wireless bus structure) or may be distributedly located (e.g., cloud computing via indirect coupling via a local area network and/or a wide area network). Further note that if the processing module, module, processing circuit, and/or processing unit implements one or more of its functions via a state machine, analog circuitry, digital circuitry, and/or logic circuitry, the memory and/or memory element storing the corresponding operational instructions may be embedded within, or external to, the circuitry comprising the state machine, analog circuitry, digital circuitry, and/or logic circuitry. Still further note that, the memory element may store, and the processing module, module, processing circuit, and/or processing unit executes, hard coded and/or operational instructions corresponding to at least some of the steps and/or functions illustrated in one or more of the Figures. Such a memory device or memory element can be included in an article of manufacture.

One or more embodiments have been described above with the aid of method steps illustrating the performance of specified functions and relationships thereof. The boundaries and sequence of these functional building blocks and method steps have been arbitrarily defined herein for convenience of description. Alternate boundaries and sequences can be defined so long as the specified functions and relationships are appropriately performed. Any such alternate boundaries or sequences are thus within the scope and spirit of the claims. Further, the boundaries of these functional building blocks have been arbitrarily defined for convenience of description. Alternate boundaries could be defined as long as the certain significant functions are appropriately performed. Similarly, flow diagram blocks may also have been arbitrarily defined herein to illustrate certain significant functionality.

To the extent used, the flow diagram block boundaries and sequence could have been defined otherwise and still perform the certain significant functionality. Such alternate definitions of both functional building blocks and flow diagram blocks and sequences are thus within the scope and spirit of the claims. One of average skill in the art will also recognize that the functional building blocks, and other illustrative blocks, modules and components herein, can be implemented as illustrated or by discrete components, application specific integrated circuits, processors executing appropriate software and the like or any combination thereof.

In addition, a flow diagram may include a “start” and/or “continue” indication. The “start” and “continue” indications reflect that the steps presented can optionally be incorporated in or otherwise used in conjunction with other routines. In this context, “start” indicates the beginning of the first step presented and may be preceded by other activities not specifically shown. Further, the “continue” indication reflects that the steps presented may be performed multiple times and/or may be succeeded by other activities not specifically shown. Further, while a flow diagram indicates a particular ordering of steps, other orderings are likewise possible provided that the principles of causality are maintained.

The one or more embodiments are used herein to illustrate one or more aspects, one or more features, one or more concepts, and/or one or more examples. A physical embodiment of an apparatus, an article of manufacture, a machine, and/or of a process may include one or more of the aspects, features, concepts, examples, etc. described with reference to one or more of the embodiments discussed herein. Further, from figure to figure, the embodiments may incorporate the same or similarly named functions, steps, modules, etc. that may use the same or different reference numbers and, as such, the functions, steps, modules, etc. may be the same or similar functions, steps, modules, etc. or different ones.

Unless specifically stated to the contra, signals to, from, and/or between elements in a figure of any of the figures presented herein may be analog or digital, continuous time or discrete time, and single-ended or differential. For instance, if a signal path is shown as a single-ended path, it also represents a differential signal path. Similarly, if a signal path is shown as a differential path, it also represents a single-ended signal path. While one or more particular architectures are described herein, other architectures can likewise be implemented that use one or more data buses not expressly shown, direct connectivity between elements, and/or indirect coupling between other elements as recognized by one of average skill in the art.

The term “module” is used in the description of one or more of the embodiments. A module implements one or more functions via a device such as a processor or other processing device or other hardware that may include or operate in association with a memory that stores operational instructions. A module may operate independently and/or in conjunction with software and/or firmware. As also used herein, a module may contain one or more sub-modules, each of which may be one or more modules.

As may further be used herein, a computer readable memory includes one or more memory elements. A memory element may be a separate memory device, multiple memory devices, or a set of memory locations within a memory device. Such a memory device may be a read-only memory, random access memory, volatile memory, non-volatile memory, static memory, dynamic memory, flash memory, cache memory, and/or any device that stores digital information. The memory device may be in a form a solid state memory, a hard drive memory, cloud memory, thumb drive, server memory, computing device memory, and/or other physical medium for storing digital information.

While particular combinations of various functions and features of the one or more embodiments have been expressly described herein, other combinations of these features and functions are likewise possible. The present disclosure is not limited by the particular examples disclosed herein and expressly incorporates these other combinations. 

What is claimed is:
 1. A method comprises: obtaining, by a computing device of a dispersed storage network (DSN), an encoding matrix that is in accordance with a dispersed storage error encoding function, wherein the encoding matrix includes a first matrix section and a second matrix section, wherein the first matrix section includes “D−1” number of rows of a unity matrix, wherein “D” is a decode threshold number of the dispersed storage error encoding function, and wherein the second matrix section includes “T−D+1” rows of encoding terms, wherein “T” is a pillar width number of the dispersed storage error encoding function; dividing, by the computing device, a data element into “Z” data element blocks, wherein “Z” is a function of “D” and a total number of data blocks of a data segment, wherein the data segment includes the data element and “D−1” random elements; placing, by the computing device, the “Z” data element blocks in a first row of a data matrix, wherein the first row corresponds to a missing row of the “D−1” number of rows of the unity matrix; dividing, by the computing device, the “D−1” random elements into a plurality of random element blocks, wherein each of the “D−1” random elements is divided into “Z” random element blocks; placing, by the computing device, the plurality of random element blocks in other rows of the data matrix; and matrix multiplying, by the computing device, the encoding matrix with the data matrix to produce a set of encoded data slices.
 2. The method of claim 1 further comprises: sending, by the computing device, the set of encoded data slices to a set of storage units of the DSN for storage therein.
 3. The method of claim 1 further comprises: placing, by the computing device, the “Z” data element blocks in a second row of the data matrix, wherein the second row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 4. The method of claim 1 further comprises: placing, by the computing device, the “Z” data element blocks in a third row of the data matrix, wherein the third row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 5. The method of claim 1, wherein the second matrix section includes “T−D+1” rows of a Vandermonde matrix.
 6. The method of claim 1, wherein a random element of the “D−1” random elements comprises one or more of: a pseudo random generated number; a known binary number pattern; and a random alpha numeric sequence.
 7. A computing device of a dispersed storage network (DSN), the computing device comprises: an interface; memory; and a processing module operably coupled to the memory and the interface, wherein the processing module is operable to: obtain an encoding matrix that is in accordance with a dispersed storage error encoding function, wherein the encoding matrix includes a first matrix section and a second matrix section, wherein the first matrix section includes “D−1” number of rows of a unity matrix, wherein “D” is a decode threshold number of the dispersed storage error encoding function, and wherein the second matrix section includes “T−D+1” rows of encoding terms, wherein “T” is a pillar width number of the dispersed storage error encoding function; divide a data element into “Z” data element blocks, wherein “Z” is a function of “D” and a total number of data blocks of a data segment, wherein the data segment includes the data element and “D−1” random elements; place the “Z” data element blocks in a first row of a data matrix, wherein the first row corresponds to a missing row of the “D−1” number of rows of the unity matrix; dividing, by the computing device, the “D−1” random elements into a plurality of random element blocks, wherein each of the “D−1” random elements is divided into “Z” random element blocks; place the plurality of random element blocks in other rows of the data matrix; and matrix multiply the encoding matrix with the data matrix to produce a set of encoded data slices.
 8. The computing device of claim 7, wherein the processing module is further operable to: send the set of encoded data slices to a set of storage units of the DSN for storage therein.
 9. The computing device of claim 7, wherein the processing module is further operable to: place the “Z” data element blocks in a second row of the data matrix, wherein the second row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 10. The computing device of claim 7, wherein the processing module is further operable to: place the “Z” data element blocks in a third row of the data matrix, wherein the third row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 11. The computing device of claim 7, wherein the second matrix section includes “T−D+1” rows of a Vandermonde matrix.
 12. The computing device of claim 7, wherein a random element of the “D−1” random elements comprises one or more of: a pseudo random generated number; a known binary number pattern; and a random alpha numeric sequence.
 13. A computer readable memory comprises: a first memory that stores operational instructions that, when executed by a computing device of a dispersed storage network (DSN), causes the computing device to: obtain an encoding matrix that is in accordance with a dispersed storage error encoding function, wherein the encoding matrix includes a first matrix section and a second matrix section, wherein the first matrix section includes “D−1” number of rows of a unity matrix, wherein “D” is a decode threshold number of the dispersed storage error encoding function, and wherein the second matrix section includes “T−D+1” rows of encoding terms, wherein “T” is a pillar width number of the dispersed storage error encoding function; divide a data element into “Z” data element blocks, wherein “Z” is a function of “D” and a total number of data blocks of a data segment, wherein the data segment includes the data element and “D−1” random elements; place the “Z” data element blocks in a first row of a data matrix, wherein the first row corresponds to a missing row of the “D−1” number of rows of the unity matrix; dividing, by the computing device, the “D−1” random elements into a plurality of random element blocks, wherein each of the “D−1” random elements is divided into “Z” random element blocks; place the plurality of random element blocks in other rows of the data matrix; and matrix multiply the encoding matrix with the data matrix to produce a set of encoded data slices.
 14. The computer readable memory of claim 13, wherein a second memory that stores operational instructions that, when executed by the computing device, causes the computing device to: send the set of encoded data slices to a set of storage units of the DSN for storage therein.
 15. The computer readable memory of claim 13, wherein the first memory further stores operational instructions that, when executed by the computing device, causes the computing device to: place the “Z” data element blocks in a second row of the data matrix, wherein the second row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 16. The computer readable memory of claim 13, wherein the first memory further stores operational instructions that, when executed by the computing device, causes the computing device to: place the “Z” data element blocks in a third row of the data matrix, wherein the third row corresponds to the missing row of the “D−1” number of rows of the unity matrix.
 17. The computer readable memory of claim 13, wherein the second matrix section includes “T−D+1” rows of a Vandermonde matrix.
 18. The computer readable memory of claim 13, wherein a random element of the “D−1” random elements comprises one or more of: a pseudo random generated number; a known binary number pattern; and a random alpha numeric sequence. 